Method of nose stretching in Newton's problem of minimal resistance

Abstract

We consider the problem $\inf \{\int {\int }_{\Omega }(1+|\nabla u(x,y){|}^2{)}^{-1}dxdy:\text{ the function }u:\Omega \to \mathbb{R}\text{ is concave and }0\le u(x,y)\le M\text{ for all }(x,y)\in \Omega =\{(x,y):x^2+y^2\le 1\}\}$ (Newton's problem) and its generalizations. In the paper \cite{BrFK} it is proved that if a solution $u$ is $C^2$ in an open set $\mathcal{U}\subset \Omega$ then $\det D^{2}u=0$ in $\mathcal{U}$. It follows that graph$(u){\rfloor }_{\mathcal{U}}$ does not contain extreme points of the subgraph of $u$. In this paper we prove a somewhat stronger result. Namely, there exists a solution $u$ possessing the following property. If $u$ is $C^1$ in an open set $\mathcal{U}\subset \Omega$ then graph$(u{\rfloor }_{\mathcal{U}})$ does not contain extreme points of the convex body $C_u=\{(x,y,z):(x,y)\in \Omega ,0\le z\le u(x,y)\}$. As a consequence, we have $C_u = \text{\rm Conv} (\overline{\text{\rm Sing$C_u$}})$, where Sing$C_u$ denotes the set of singular points of $\partial C_u$. We prove a similar result for a generalized Newton's problem.

28 pages, 5 figures

References in corpus (4)

• Newton's aerodynamic problem in media of chaotically moving particles
• Spinning rough disk moving in a rarefied medium
• The problem of camouflaging via mirror reflections
• Behavior of convex surfaces near ridge points